MathsCBSE Class 10 · Standard · Board exam 2025, set 30/3/1 · 5 marks · Long answer

CBSE Class 10 Mathematics Standard 2025: question 35

Set in the CBSE Class 10 board exam of 2025, paper 30/3/1, section D, question 35, for 5 marks. Try it first; the hints and our step-by-step solution are below.

Q35 · 5 marks · Long answerSurface Areas and Volumes

A vessel is in the form of an inverted cone. Its height is 8 cm and the radius of its top, which is open, is 5 cm. It is filled with water up to the brim. When lead shots, each of which is a sphere of radius $0 \cdot 5$ cm, are dropped into the vessel, one-fourth of the water flows out. Find the number of lead shots dropped in the vessel.

Source: official question paper from cbse.gov.in (PDF). See it in the full 2025 board paper, set 30/3/1.

Hints

Open one hint at a time and try again before the next.

Hint 1

The water that flows out takes up exactly the same space as the lead shots.

Hint 2

Find the cone's volume, take one-fourth of it, then divide by the volume of one sphere of radius 0.5 cm.

Step-by-step solution

Each step shows the marks it earns, as an examiner would award them. Checked by an independent re-solve from the question paper.

Cone: $r = 5$ cm, $h = 8$ cm. Volume of water in the full cone $= \frac{1}{3} π r^{2} h = \frac{1}{3} π \times 5^{2} \times 8$
1 mark
$= \frac{200 π}{3}$ cm $^{3}$
½ mark
The water that flows out equals the total volume of the lead shots dropped in. Volume of water that flows out $= \frac{1}{4} \times \frac{200 π}{3} = \frac{50 π}{3}$ cm $^{3}$
1 mark
Radius of one lead shot $= 0.5 = \frac{1}{2}$ cm. Volume of one shot $= \frac{4}{3} π r^{3} = \frac{4}{3} π (\frac{1}{2})^{3}$
1 mark
$= \frac{π}{6}$ cm $^{3}$
½ mark
Number of lead shots $= \frac{50 π /3}{π /6} = \frac{50 π}{3} \times \frac{6}{π} = 100$
1 mark

Answer: 100 lead shots

Where the marks are

Writes the volume of the cone as $\frac{1}{3} π (5)^{2} (8)$ (1 mark)
Evaluates it as $\frac{200 π}{3}$ cm $^{3}$ (½ mark)
Finds the water displaced as $\frac{50 π}{3}$ cm $^{3}$ (1 mark)
Writes the volume of one shot as $\frac{4}{3} π (0.5)^{3}$ (1 mark)
Evaluates it as $\frac{π}{6}$ cm $^{3}$ (½ mark)
Divides the two volumes to get 100 shots (1 mark)

Revise the chapter

Chapter: Surface Areas and Volumes (notes, formulas, where marks are lost and every past question)

Practise this chapterBoard-style questions from Surface Areas and Volumes, one at a time, marked against the answer key and marking points.Back to the full paper2025 board paper, set 30/3/1 (Standard): every question, by section.

The question is transcribed from the official paper; figures are described in words. The solution, hints and chapter tag are ours. Spotted an error? Write tosupport@questionbuddy.in.

CBSE Class 10 Mathematics Standard 2025: question 35

Hints

Step-by-step solution

Revise the chapter

More questions from Surface Areas and Volumes